Journal of Knot Theory and Its Ramifications

The ordinary unknotting number of 1-dimensional knots, which is defined by using the crossing-change, is a very basic and important invariant. Let $n$ be a positive integer. It is very natural to consider the “unknotting number” associated with other local moves on $n$-dimensional knots. In this paper, we prove the following. For the ribbon-move on 2-knots, which is a local move on knots, we have the following: There is a 2-knot which is changed into the unknot by two times of the ribbon-move not by one time. The “unknotting number” associated with the ribbon-move is unbounded. For the pass-move on 1-knots, which is a local move on knots, we have the following: There is a 1-knot such that it is changed into the unknot by two times of the pass-move not by one time and such that the ordinary unknotting number is $4$. For any positive integer $n$, there is a 1-knot whose “unknotting number” associated with the pass-move is $>n$ and whose ordinary unknotting number is $4n$. Let $p$ and $q$ be positive integers. For the $(p,q)$-move on $(p+q-1)$-knots, which is a local move on knots, we have the following: Let $k$ be a non-negative integer. There is a $(4k+2)$-knot which is changed into the unknot by two times of the $(2k+1,2k+2)$-move not by one time. The “unknotting number” associated with the $(2k+1,2k+2)$-move is unbounded. There is a $(4k+1)$-knot which is changed into the unknot by two times of the $(2k+1,2k+1)$-move not by one. The “unknotting number” associated with the $(2k+1,2k+1)$-move is unbounded. We prove the following: For any positive integer $n$ and any positive integer $m<n$, there is a $(4k+1)$-knot which is changed into the unknot by $n$ times of the twist-move not by $m$ times.

It is well known that surface-links in $4$-space can be presented by diagrams on the plane of $4$-valent spatial graphs with makers on the vertices, called marked graph diagrams. In this paper, we extend the method of presenting surface-links by marked graph diagrams to presenting immersed surface-links. We also give some moves on marked graph diagrams that preserve the ambient isotopy classes of their presenting immersed surface-links.

We show that the definition of unrolled Hopf algebras can be naturally extended to the Nichols algebra ${ℬ}(V)$ of a Yetter–Drinfeld module $V$ on which a Lie algebra $𝔤$ acts by biderivations. As a special case, we find unrolled versions of the small quantum group.

For a signed cyclic graph $G$, we can construct a unique virtual link $L$ by taking the medial construction and converting 4-valent vertices of the medial graph to crossings according to the signs. If a virtual link can occur in this way then we say that the virtual link is graphical. In this paper, we shall prove that a virtual link $L$ is graphical if and only if it is checkerboard colorable. On the other hand, we introduce a polynomial $F[G]$ for signed cyclic graphs, which is defined via a deletion-marking recursion. We shall establish the relationship between $F[G]$ of a signed cyclic graph $G$ and the bracket polynomial of one of the virtual link diagrams associated with $G$. Finally, we give a spanning subgraph expansion for $F[G]$.

Knot mosaics are used to model physical quantum states. The mosaic number of a knot is the smallest integer $m$ such that the knot can be represented as a knot $m$-mosaic. In this paper, we establish an upper bound for the crossing number of a knot in terms of the mosaic number. Given an $m$-mosaic and any knot $K$ that is represented on the mosaic, its crossing number $c$ is bounded above by ${(m-2)}^2-2$ if $m$ is odd, and by ${(m-2)}^2-(m-3)$ if $m$ is even. In the process, we develop a useful new tool called the mosaic complement.

In [K. Kishimoto, T. Shibuya and T. Tsukamoto, Simple-ribbon fusions and genera of links, J. Math. Soc. Japan 68 (2016) 1033–1045], we introduced a special kind of fusions, the simple-ribbon fusions. In this paper, we give a sufficient condition for simple-ribbon fusions to produce a prime knot.

A $2$-knot is a surface in $R^4$ that is homeomorphic to $S^2$, the standard sphere in $3$-space. A ribbon $2$-knot is a $2$-knot obtained from $m$ $2$-spheres in $R^4$ by connecting them with $m-1$ pipes. Let $K^2$ be a ribbon 2-knot. The ribbon crossing number, denoted by $r$-$cr(K^2)$, is a numerical invariant of the ribbon $2$-knot $K^2$. In [T. Yasuda, Crossing and base numbers of ribbon 2-knots, J. Knot Theory Ramifications 10 (2001) 999–1003] we showed that there exist just $17$ ribbon $2$-knots of the ribbon crossing number up to three. In this paper, we show that there exist no more than $111$ ribbon $2$-knots of ribbon crossing number four.

A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell and Michael, and, independently, Mattman, showed that intrinsically knotted graphs have at least 21 edges. Recently, Lee, Kim, Lee and Oh, and, independently, Barsotti and Mattman, showed that $K_7$ and the 13 graphs obtained from $K_7$ by $\nabla Y$ moves are the only intrinsically knotted graphs with 21 edges. Also Kim, Lee, Lee, Mattman and Oh showed that there are exactly three triangle-free intrinsically knotted graphs with 22 edges having at least two vertices of degree 5. Furthermore, there is no triangle-free intrinsically knotted graph with 22 edges that has a vertex with degree larger than 5. In this paper, we show that there are exactly five triangle-free intrinsically knotted graphs with 22 edges having exactly one degree 5 vertex. These are Cousin 29 of the $K_{3,3,1,1}$ family, Cousins 97 and 99 of the $E_9+e$ family and two others that were previously unknown.